Insider Trading With a Random Deadlinepeople.stern.nyu.edu/rcaldent/papers/RInsider.pdf · insider could let the information asymmetry (the variance of the uninformed traders’ estimate

http://www.econometricsociety.org/

Econometrica, Vol. 78, No. 1 (January, 2010), 245–283

INSIDER TRADING WITH A RANDOM DEADLINE

RENÉ CALDENTEYStern School of Business, New York University, New York, NY 10012, U.S.A.

ENNIO STACCHETTINew York University, New York, NY 10012, U.S.A.

The copyright to this Article is held by the Econometric Society. It may be downloaded,printed and reproduced only for educational or research purposes, including use in coursepacks. No downloading or copying may be done for any commercial purpose without theexplicit permission of the Econometric Society. For such commercial purposes contactthe Office of the Econometric Society (contact information may be found at the websitehttp://www.econometricsociety.org or in the back cover of Econometrica). This statement mustbe included on all copies of this Article that are made available electronically or in any otherformat.


Econometrica, Vol. 78, No. 1 (January, 2010), 245–283

INSIDER TRADING WITH A RANDOM DEADLINE

BY RENÉ CALDENTEY AND ENNIO STACCHETTI1

We consider a model of strategic trading with asymmetric information of an assetwhose value follows a Brownian motion. An insider continuously observes a signal thattracks the evolution of the asset’s fundamental value. The value of the asset is publiclyrevealed at a random time. The equilibrium has two regimes separated by an endoge-nously determined time T . In [0�T ), the insider gradually transfers her information tothe market. By time T , all her information has been transferred and the price agreeswith the market value of the asset. In the interval [T�∞), the insider trades large vol-umes and reveals her information immediately, so market prices track the market valueperfectly. Despite this market efficiency, the insider is able to collect strictly positiverents after T .

KEYWORDS: Insider trading, Kyle model, market microstructure, asset pricing.

1. INTRODUCTION

THIS PAPER STUDIES a model of strategic trading with asymmetric informationof an asset whose value follows a Brownian motion. An insider receives a flowof (noisy) signals that tracks the evolution of the asset value. Other tradersreceive no signals and can only observe the total volume of trade. Since thereis uncertainty about the value of the asset before the game starts, the first signalgenerates a lumpy informational asymmetry between the insider and the restof the market participants. Subsequently, the insider receives a sequence ofupdates regarding the fundamental valuation of the asset. At an unpredictabletime, a public announcement reveals the current value of the asset to all thetraders. In equilibrium, the insider releases all her private information by afinite time T and keeps the market fully informed thereafter. Thus, she doesnot find it profitable to maintain informational asymmetry indefinitely.

Kyle (1985) introduced a dynamic model of insider trading2 where an insiderreceives only one signal and the fundamental asset value does not change overtime. Through trade, the insider progressively releases her private informationto the market as she exploits her informational advantage. The market is alsopopulated by many liquidity traders who are uninformed and trade randomly.At time 0, the insider observes the value of an asset. The same information ispublicly released later, at time 1, to all market participants. In each trading pe-riod in the time interval [0�1], traders submit order quantities to a risk-neutralmarket maker who sets prices competitively and trades in his own account toclear the market. The market maker cannot observe individual trades, but can

1We gratefully acknowledge the feedback of David Pearce. We also thanks Markus Brunner-meier, Lasse Pedersen, and Debraj Ray and seminar participants at NYU.

2Glosten and Milgrom (1985) proposed an alternative formalization of Bagehot’s (1971) in-formal model.

© 2010 The Econometric Society DOI: 10.3982/ECTA7884



http://dx.doi.org/10.3982/ECTA7884

246 R. CALDENTEY AND E. STACCHETTI

observe the total volume of trade in each trading period. The market makeralso knows (in equilibrium) the strategy of the informed trader, and sets pricesefficiently, conditional on past and present volumes of trade.

Kyle constructed a linear equilibrium where in each period the price adjust-ment is proportional to the volume of trade, and the insider’s volume of trade isproportional to the gap between the asset value and the current market price.The market maker’s estimate of the asset value, reflected in the current marketprice, improves over time. As the public announcement date approaches, thisestimate converges to the value of the asset and the insider trades franticallyin her desire to exploit any price differential.

Our model differs from Kyle’s model in three important ways. First, the fun-damental value of the asset follows a Brownian motion and, therefore, changescontinuously over time. Second, in addition to the initial observation, the in-sider continuously receives a signal of the current fundamental value of theasset. Third, the public announcement date is unpredictable: it has an expo-nential distribution.

The first difference by itself is irrelevant. In Kyle’s model it makes no dif-ference whether at time 0 the insider observes the true value of the asset orjust an unbiased signal. Moreover, the model where the insider observes thetrue value and the value of the asset follows a Brownian motion is formallyequivalent to a model where the initial observation is an unbiased signal ofthe final value of the asset. But this feature of our model becomes importantwhen it is combined with the second feature. Finally, the third feature removesthe pressure in Kyle’s model behind the trade frenzy that occurs as the an-nouncement date approaches. In our model, where the announcement date isnot deterministic, the insider has no urgency to exhaust all arbitrage oppor-tunities, and release all her private information in the process, by a particulardeadline. Thus, while it is evident that in Kyle’s model the price will becomeefficient (in the sense that it incorporates all the available information) as timereaches the announcement date, it is unclear whether in our model the insiderwill ever fully reveal her private information.

It is exactly this feature of the equilibrium of the fixed horizon model thatBack (1992) exploited to develop his elegant “backward programming” solu-tion method. In a model with a random horizon, Back’s method is not directlyapplicable.

Our model is not the first to introduce a public announcement with ran-dom time. Back and Baruch (2004) compared the models of Kyle (1985) andGlosten and Milgrom (1985). To facilitate the comparison, they adopted aGlosten and Milgrom model with a single long-lived insider (who times hertransactions strategically) and a Kyle model with a random terminal time anda risky asset that takes only the values 0 or 1.

Kyle’s original model is in discrete time. However, Kyle also showed that asthe period length Δ converges to 0, the equilibrium converges to an equilibriumof the continuous-time limit model. He then interpreted the continuous-time

INSIDER TRADING WITH A RANDOM DEADLINE 247

model as a good representation of a discrete-time model where the agents cantrade frequently. We maintain this interpretation and view the continuous-timemodel as a mathematical convenience that affords us the powerful tools of sto-chastic calculus. The discrete-time version of our model has a unique equilib-rium that converges to a well defined strategy profile as Δ ↓ 0. However, in ourcase, this limit strategy is not an equilibrium of the continuous-time model. Theinterpretation of the continuous-time model is therefore delicate and needs tobe examined more carefully. The lack of “continuity” arises because in thelimit the insider wants to trade at infinite rates after some time T . She stillcollects positive rents after T even though the price perfectly tracks the valueof the asset. However, after T , her payoff function evaluated at the limit strat-egy is 0. Therefore, as we explain in Section 5, the limit strategy cannot be anequilibrium of the continuous-time model. Because our characterization of theequilibrium has a crisper form in the limit, our discussion below refers to thelimit equilibrium.

Our model includes various special cases. The value of the asset remainsconstant over time if the variance of its Brownian motion is reduced to 0. Sincein our model the insider observes the initial value without noise, the signalsthat track the value of the asset over time become superfluous. This version ofour model is similar to Kyle’s model, where the insider is endowed only with aninitial piece of private information, but with a random end time. Alternatively,we can specialize our model to give the insider no initial informational advan-tage. This is accomplished by informing all traders of the initial value of theasset. In this version of the model, the insider’s informational advantage arisesexclusively from her ability to observe the evolution of the asset value. Thisis an important model in its own right. An interesting question in this modelis how the insider “manages” the information asymmetry. For example, theinsider could let the information asymmetry (the variance of the uninformedtraders’ estimate of the current value) grow to reach asymptotically a certainlimit or grow without bound. The larger is the information asymmetry, themore likely it is that the market will substantially misprice the asset and, there-fore, the larger are the profitable arbitrage opportunities. Thus, in this modelas well it is not evident how much of the insider’s information is incorporatedin the market price and how quickly this happens. We study this special casein the process of constructing an equilibrium for our general model. It turnsout that in equilibrium the insider fully reveals her information as soon as shereceives it. Hence, the market price equals the asset value at all times. Yet, theinsider makes strictly positive profits. In independent work, Chau and Vayanos(2008) reached the same conclusion (for this case without initial informationalasymmetry) in a slightly different model. They assumed that the insider re-ceives a flow of information, the asset pays a dividend, and there is no publicannouncement. In addition, they assumed that the market maker continuouslyobserves a noisy signal of the value of the asset. In the absence of this noisysignal, their model would be formally equivalent to ours. Chau and Vayanos


(2008) limited attention to the steady state of their model and did not studyhow the equilibrium approaches the steady state. One implication of our re-sults is that in the absence of an initial information asymmetry, the steady stateis reached “immediately” (as the period length goes to 0), so although Chauand Vayanos (2008) assumed that trading had been taking place indefinitely,this is not needed.

The equilibrium of our general model has a remarkable feature. There is atime T , endogenously determined in equilibrium, by which the insider revealsall her information (if the public announcement has not yet occurred). Thus,even though there is no deterministic deadline, the price converges to the assetvalue at time T . Moreover, time T divides the equilibrium into two phases. Aslong as the public announcement does not occur, in the interval [0�T ) the in-sider gradually transfers her information to the market and the market’s uncer-tainty about the value of the asset decreases to 0 monotonically. In the interval[T�∞), the insider trades large volumes and reveals her information imme-diately, so market prices track the asset value perfectly. Nevertheless, as weexplained above, after T , the insider collects strictly positive rents.

There is a vast literature on insider trading3 and many papers have extendedKyle’s model. Holden and Subrahmanyam (1992) and Foster and Viswanathan(1996) considered a market with multiple competing insiders. They showedthat competition among insiders accelerates the release of their private infor-mation. In a one-period model with heterogeneous insiders, Spiegel and Sub-rahmanyam (1992) replaced Kyle’s uninformed liquidity traders with strategicutility-maximizing agents trading for hedging purposes. In a multiperiod set-ting, Mendelson and Tunca (2004) proposed an alternative endogenous liquid-ity trading model that allowed for various types of market information. Similarto our model, Back and Pedersen (1998) considered the case where the in-sider continuously observes private information. To prove that an equilibriumexists, they assumed that the insider’s initial amount of private informationis relatively high compared to the flow of new information and that this flowdecreases fast over time. We show that a similar condition is required in thecontinuous-time version of our model. Furthermore, we also show that it isprecisely when this condition is violated—that is, when the insider’s initial pri-vate information is small compared to the inflow on new information—thatour equilibrium reaches market efficiency at a fixed time T .

The rest of the paper is organized as follows. Section 2 introduces thediscrete-time model and Section 3 constructs a Markovian equilibrium. In Sec-tion 4, we show that the Markovian equilibrium has a well defined limit equi-librium as the period length goes to 0 and we provide a full characterizationof it. In Section 5, we formulate the continuous-time model and show that the

3For a comprehensive review of this literature and its connection to the broader market mi-crostructure theory, we refer the reader to O’Hara (1997), Brunnermeier (2001), Biais, Glosten,and Spatt (2005), Amihud, Mendelson, and Pedersen (2006), and references therein.


limit equilibrium is not an equilibrium. Section 6 includes our concluding re-marks.

2. MODEL DESCRIPTION

The market participants are the insider, the market maker, and a (large)number of liquidity traders. The market maker opens the floor for trading onlyat discrete times {tn}n≥0. These trading dates are evenly spaced over time (e.g.,once a day) so that tn = nΔ for some positive constant Δ. The interval of time[tn� tn+1) is called period n. During period n, the following sequence of eventsoccurs. First, the insider (and only her) receives private information about thefundamental value Vn of the asset. Then the insider and the liquidity traderssimultaneously place buy/sell orders xn and yn, respectively, for a quantity ofthe asset. An order is a binding contract to buy/sell a quantity of the asset(the “size of the order”) at a price determined by the market maker. Finally,after observing the total volume of trade zn = xn + yn, the market maker setsthe price pn and trades the necessary quantity to close all orders. We assumethat the market maker is not able to differentiate between insider and liquiditytrading. He only observes the net volume of trade zn.

This trading process continues until a random time τ, independent of thehistory of transactions and prices, when the fundamental value of the asset be-comes public knowledge. At this time, the market price immediately matchesthe fundamental value and the insider loses her informational advantage. Weassume that the public announcement occurs at the end of a period (after trad-ing). That is, τ = ηΔ, where η≥ 0 has a geometric distribution with probabilityof failure ρ= e−μΔ for some fixed μ> 0.

Liquidity traders are not strategic agents and they trade for idiosyncraticreasons. In particular, we assume that {yn}n≥0 is a sequence of independentand identically distributed (i.i.d.) normal random variables with mean 0 andvariance Σy = σ2

y Δ. On the other hand, the insider trades strategically so as tomaximize her expected net payoff during [0� τ]. The insider’s payoff is drivenby her informational advantage as she alone observes the evolution of the fun-damental value of the asset Vn during [0� τ]. We assume that Vn evolves as arandom walk {Vn = Vn−1 + vn}n≥1, where V0 is normally distributed with meanV0 and variance Σ0, and {vn}n≥1 is a sequence of i.i.d. normal random variableswith mean 0 and variance Σv = σ2

vΔ. The market maker and the rest of themarket participants only know the distributions of V0 and {vn}n≥1. Hence, V0

represents a lumpy endowment of private information that the insider gets attime 0, while {vn}n≥1 represents the incremental private information that shereceives over time.

At the beginning of each period n, before the fundamental value becomespublic knowledge, the market maker commits to a pricing rule (that is legallybinding). The rule specifies the price pn for the current period’s transactions asa function of the total volume of trade zn and the public history up to this time.


(For completeness, we define p−1 = E[V0] = V0.) The insider and the liquiditytraders place their orders after the rule is announced. All orders are executedat the end of the period. The market maker observes the public history of pricesand (total) volumes of trade. His information in period n is represented by thehistory FM

n = (z0�p0� � zn−1�pn−1� zn). Similarly, the insider’s information in-cludes the public history of prices and trades, the private history of her orders,and the fundamental values she has observed. Her information in period nis represented by the history F I

n = (v0�x0� z0�p0� � xn−1� zn−1�pn−1� vn). Theinsider places her order at the beginning of the period, after observing the cur-rent value of the fundamental.

The insider and the market maker are risk-neutral agents. Given a trajectoryX = {xn} for the insider’s trading and P = {pn} for market prices, the insider’spayoff is

Π(P�X)=η∑n=0

[Vη −pn]xn

The insider maximizes the expected value ofΠ(P�X). Since η has a geometricdistribution,

E[Π(P�X)] = E

[ ∞∑n=0

ρn[Vn −pn]xn]

DEFINITION 1: A strategy for the market maker is an FMn -adapted process

P = {pn}0≤n≤η, and a strategy for the insider is an F In -adapted process X =

{xn}0≤n≤η. The profile (P�X) is an equilibrium if (i) for any n≥ 0

pn = E[Vn |X�FMn ]�

and (ii) given P , X maximizes E[Π(P�X)]. Here E[Vn|X�FMn ] means the con-

ditional expectation of Vn given the public history FMn at time n and the in-

sider’s strategy X , which specifies how she trades every period as a function ofher information.

We do not model explicitly competition among market makers, but we im-plicitly assume that our market maker competes in prices with other marketmakers.4 In equilibrium, this competition drives the market maker to set the

4The model is not exactly a game and our definition of an equilibrium differs from that of aNash equilibrium. However, Kyle (1985) suggested that the two definitions would coincide in agame where two market makers simultaneously bid prices after observing the current volume oftrade and the winner gets the right to clear the market at the winning price. To avoid collusion,we can assume that there is a large population of market makers and that each market makerparticipates in the bidding game only once.


price equal to the expected value of the asset’s fundamental value given thehistory of information he has observed and the insider’s trading strategy (thisis condition (i) in Definition 1). The insider chooses her strategy so as to max-imize her expected discounted profit, given that she knows how the marketmaker will choose prices (this is condition (ii) in Definition 1).

We will restrict attention to Markovian equilibria with a particular statespace. At the beginning of period n, before the market maker observes thevolume of trade, the state is (n� Vn�Σn), where Vn = E[Vn|FM

n−1�X] is the mar-ket maker’s estimate of Vn and Σn = E[(Vn − Vn)

2|FMn−1�X] is the variance

of this estimate, conditional on the insider’s strategy X and market infor-mation available at the end of period n − 1. Note that since the marketmaker’s estimate of Vn depends on the strategy X of the insider, the stateand corresponding Markovian strategy profile need to be specified simulta-neously.

DEFINITION 2: A strategy profile (P�X) is Markovian if for each n, the in-sider’s order xn and the market maker’s price pn depend only on the currentstate (n� Vn�Σn) and the signals they receive in period n: vn for the insiderand zn for the market maker. In this case, we write xn =X(n� Vn�Σn�Vn) andpn = P(n� Vn�Σn� zn). If (P�X) is a Markovian strategy profile, let

Πn(Vn�Σn�Vn)= E

[η∑k=n(Vk −pk)xk

∣∣∣Vn�Σn�Vn� (P�X)]

be the insider’s expected payoff-to-go for the transactions made from periodn until the fundamental value is publicly revealed, when the current state is(n� Vn�Σn) and the insider observes Vn. When (P�X) is a Markovian equilib-rium, pn−1 = Vn for all n.

3. MARKOVIAN EQUILIBRIUM

In this section, we construct a linear Markovian equilibrium (P�X), that is, aMarkovian equilibrium such that

P(n� Vn�Σn� zn)= Vn + λn(Σn)zn�(1)

X(n� Vn�Σn�Vn)= βn(Σn)(Vn − Vn)�where {λn} and {βn} are sequences of functions λn�βn : R++ → R+. The con-struction exploits the key property that the trajectory {Σn} is deterministic andindependent of the history of trades. As a result, the sequences {λn} and {βn}are also deterministic and hereafter we drop the arguments Σn (we also dropthis argument in the function Πn). Moreover, since in equilibrium Vn = pn−1

for all n, hereafter we do not differentiate these two variables and write, for


example, Πn(pn−1� Vn) instead of Πn(Vn�Σn�Vn).

THEOREM 1: There exist unique sequences {λn}� {βn} ∈ R++ such that the lin-ear strategy profile (P�X) defined by (1) is a Markovian equilibrium. In equi-librium, {Σn} is a deterministic trajectory that is not affected by the (stochastic)choices of the insider and the market maker. Furthermore, there exist determin-istic sequences {αn}� {γn} ⊂ R++ such that the insider’s expected payoff-to-go for(P�X) satisfies

Πn(p�V )= αn(V −p)2 + γn for all n≥ 0(2)

Given Σ0 > 0, there is a unique nonnegative value β0 that generates the equi-librium profile {(Σn�βn�λn�αn�γn)}n≥0 recursively through the systems of equa-tions

Σn+1 = Σv + ΣnΣy

β2nΣn +Σy � βn+1Σn+1 = ρβnΣn

[Σ2y

Σ2y −β4

nΣ2n

]�(3)

λn = βnΣn

β2nΣn +Σy �

αn = 1 − λnβn2λn

� ργn+1 = γn − 1 − 2λnβn2λn(1 − λnβn)(Σv + λ2

nΣy)�(4)

where

γ0 =∞∑k=0

ρk(

1 − 2λkβk2λk(1 − λkβk)

)(Σv + λ2

kΣy)(5)

PROOF: Let us consider first the market maker’s equilibrium condition(i.e., condition (i) in Definition 1). If the insider uses the trading strategyxn = βn(Vn−pn−1) for some deterministic sequence {βn}, then the market pricein period n satisfies

pn = E[Vn|zn = yn +βn(Vn −pn−1)�FMn−1]

Conditional on the available market information FMn−1, the pair (Vn� zn) is a

normally distributed two-dimensional random vector. Hence, by the projectiontheorem for normal random variables, we get that

pn = E[Vn|FMn−1] + Cov[Vn� zn|FM

n−1]Var[zn|FM

n−1](zn − E[zn|FM

n−1])

= pn−1 + βnΣn

β2nΣn +Σy zn


Note that this pricing rule satisfies condition (1) with λn as in equation (3). Inaddition,

Σn+1 = Var[Vn+1|zn�FMn−1] = Σv + Var[Vn|zn�FM

n−1]

= Σv + Var[Vn|FMn−1]

(1 − Cov[Vn� zn|FM

n−1]2

Var[Vn|FMn−1]Var[zn|FM

n−1])

= Σv + ΣnΣy

β2nΣn +Σy

Since Σn+1 is independent of zn, it follows that the sequence {Σn} is indeeddeterministic.

Let us now turn to the insider optimization problem in period n. Assume thatthe market maker uses the pricing rule pn = pn−1 +λnzn for some constant λn.Furthermore, suppose that there exist two constants αn+1 and γn+1 such thatΠn+1(p�V ) = αn+1(V − p)2 + γn+1. Then, the insider’s expected payoff-to-goin period n, Πn(pn−1� Vn), is

maxxn

E[(Vn −pn−1 − λn(xn + yn))xn

+ ρ(αn+1[Vn + vn −pn−1 − λn(xn + yn)]2 + γn+1

)]

Under the condition ρλnαn+1 < 1 (otherwise the insider’s payoff would be un-bounded), the optimization problem above is concave in x and the optimalsolution is obtained from the first-order condition

xn = βn(Vn −pn−1)� where βn = 1 − 2ρλnαn+1

2λn(1 − ρλnαn+1)(6)

Thus Xn defined by (1) is indeed the insider’s best reply function. Pluggingback the optimal value of xn into the optimization above, we get that

Πn(pn−1� Vn)= (Vn −pn−1)2

4λn(1 − ρλnαn+1)+ ρ(αn+1(Σv + λ2

nΣy)+ γn+1)

That is, Πn(pn−1� Vn) is a quadratic function of the price gap Vn − pn−1, asrequired, and the coefficients of Πn(p�V ) satisfy the recursive equationsαn = [4λn(1 − ρλnαn+1)]−1 and γn = ρ(γn+1 + αn+1(Σv + λ2

nΣy)). Use (6) andthen replace the expression for λn in (3) to obtain

αn = 14λn(1 − ρλnαn+1)

= 1 − λnβn2λn

= Σy

2βnΣn

Invert (6) and then replace the expression for λn in (3) to obtain

αn+1 = 1 − 2λnβn2ρλn(1 − λnβn) = Σ2

y −β4nΣ

2n

2ρβnΣnΣy= Σy

2βn+1Σn+1


The last equality produces the equation for βn+1Σn+1 in (3).Note that the first two equations in (3) and (4) form an independent differ-

ence equation in (Σn�βn) alone. For any initial value vector (Σ0�β0), this dif-ference equation has a unique solution. The value of Σ0 is given, but the valueof β0 is “free.” The next two equations are static: the variables (λn�αn) can becomputed independently once the sequence {(Σk�βk)} has been constructed.Similarly, given {(Σk�βk)}, {γk} is uniquely defined by the initial value γ0 andits linear dynamic equation in (4).

To complete the proof, we need to show that there exists a unique valuefor β0 that generates—through the recursions (3) and (4)—a sequence{(Σn�βn�λn�αn�γn)}n≥0 that specifies an equilibrium. In the Appendix, weshow that there exists a function Ψ : R+ → R+ such that (i) β0 > Ψ(Σ0) leadsto bluffing and (ii) β0 <Ψ(Σ0) leads to unbounded payoffs. As we discuss be-low, both bluffing and unbounded payoffs are not consistent with equilibrium.Hence, for each Σ0 > 0, β0 =Ψ(Σ0) is the only feasible choice.

Bluffing: If β0 > Ψ(Σ0), then eventually βn < 0 for some n (see the Ap-pendix). That is, the insider bluffs trading on the wrong side of the spread.Moreover, when βn < 0, (3) and (4) imply that λn < 0 and αn < 0, which is acontradiction.

Unbounded Payoffs: If β0 < Ψ(Σ0), the sequence {(Σn�βn)} converges to(∞�0). Then the market maker’s strategy specified by the corresponding se-quence {λn} allows the insider to extract unbounded payoffs (see Lemmas 1and 3 in the Appendix). But the sequence {βn} generated by (3) and (4) gen-erates a bounded payoff (see Lemma 2). Therefore, {βn} is not optimal forthe insider, despite the fact that it was constructed to satisfy local optimalityconditions. Thus, in this case, ({λn}� {βn}) is not an equilibrium.

Iterate the recursive equation for γn in (4) to get

ρnγn = γ0 −n−1∑k=0

ρk(

1 − 2λkβk2λk(1 − λkβk)

)(Σv + λ2

kΣy)

= γ0 −n∑k=1

ρkαk(Σv + λ2k−1Σy)

When β0 =Ψ(Σ0), the sequence {(Σk�βk�λk�αk)} converges (and is indepen-dent of γ0). Therefore, if we choose γ0 as in (5), then limn→∞ ρnγn = 0. Thisimplies that

limn→∞

ρnE[Πn(pn−1� Vn)] = limn→∞

ρn(αnΣn + γn)= 0

That is, the transversality condition is satisfied. Thus, the sequence {βn} is op-timal against {λn} and generates the continuation value functions Πn(p�V )=αn(V −p)2 + γn.


The optimality conditions included in (3) assume that the value function isquadratic. Lemmas 1 and 2 in the Appendix imply that for any deterministicequilibrium, the value function is quadratic. Thus, we conclude that there isa unique deterministic equilibrium ({λn}� {βn}), determined by equations (3)and (4) and the initial condition β0 =Ψ(Σ0). Q.E.D.

In general, we can characterize the set of linear Markovian equilibriaas those sequences {(Σn�βn�λn�αn�γn)}n≥0 that satisfy the recursions in (3)and (4) and converge to the (unique) stationary equilibrium of the game(Σ� β� λ� α� γ) given by

Σ= 1 +√1 − ρ√

1 − ρ Σv� β=(

Σy(1 − ρ)Σv(1 +√

1 − ρ))1/2

�(7)

λ=(Σv(1 −√

1 − ρ)Σyρ

)1/2

�

α= 12

(Σy

Σv(1 +√1 − ρ)

)1/2

� γ = ρα(Σv + λ2Σy)

1 − ρ

The following proposition highlights some additional properties of an equi-librium profile {(Σn�βn)}n≥0 and its proof follows directly from the propertiesof Ψ discussed in the proof of Theorem 1 in the Appendix. Some of theseproperties are used in the proof of Theorem 2 that characterizes the limitingcontinuous-time profile as Δ goes to 0.

PROPOSITION 1: Let {(Σn�βn)}n≥0 be the discrete-time equilibrium of Theo-rem 1. Suppose Σ0 is greater (less) than or equal to Σ. Then the sequences {Σn}and {βnΣn} are decreasing (increasing) in n.

According to Proposition 1, despite the fact that the insider’s trades are in-formative and reduce the market uncertainty, when the initial variance Σ0 < Σ,Σn increases with n. In this case, the variance reduction induced by insider trad-ing is insufficient to compensate for the additional uncertainty generated by theevolution of {Vn}, and market prices are less informative over time.

4. CONTINUOUS-TIME APPROXIMATION

In this section, we analyze the limit of the discrete-time linear equilibrium ofTheorem 1 as Δ goes to 0. In particular, we will show that the discrete-timelinear Markovian equilibrium {(Σn�βn�λn�αn�γn) :n ≥ 0} converges point-


wise to a continuous-time profile {(Σt�βt�λt�αt� γt) : t ∈ R+} in an appropriatesense.

First, let us explicitly rewrite the discrete-time model in terms of the calendartime t. For any time t ≥ 0, the corresponding trading period is denoted bynt = t/Δ. We would like to express the insider’s strategy xn = βn(Vn − pn−1)in terms of her trading rate per unit of time. For this, we define βΔ(t)= βnt /Δ.For any t ≥ 0, define the continuous time extensions

pΔ(t)= pnt−1� V Δ(t)= Vnt � ΣΔ(t)= Σnt �λΔ(t)= λnt � αΔ(t)= αnt � γΔ(t)= γnt �ΠΔ(t)=Πnt (p

Δ(t)�V Δ(t))= αΔ(t)(V Δ(t)−pΔ(t))2 + γΔ(t)�and the cumulative trading processes

XΔ(t)=nt∑k=0

xk� YΔ(t)=nt∑k=0

yk� ZΔ(t)=XΔ(t)+YΔ(t)

For ease of exposition, we assume that there exist two independent Brownianmotions Byt and Bvt such that yn = σy(By(n+1)Δ −BynΔ) and vn = σv(Bv(n+1)Δ −BvnΔ).It follows that YΔ(t) and V Δ(t) converge uniformly over compact sets to Yt =σyB

yt and Vt = σvBvt , respectively. Also, in the limit, as Δ ↓ 0, τ is exponentially

distributed with rate μ. Finally, recall that Σy = σ2y Δ, Σv = σ2

vΔ, and ρ= e−μΔ.

THEOREM 2: Let T be the unique nonnegative root of the equation

Σ0 + σ2v T = σ2

v

[e2μT − 1

2μ

]and define, for all t ≥ 0,

Σt = σ2v

2μ[e2μ(T−t)+ − 2μ(T − t)+ − 1

]� βt = σvσye

μ(T−t)+

Σt�(8)

λt = σveμ(T−t)+

σy� αt = σye

−μ(T−t)+

2σv�

γt = σyσve−μ(T−t)+

4μ[e2μ(T−t)+ + 2μ(T − t)+ + 3

]

Then (ΣΔ(t)�βΔ(t)�λΔ(t)�αΔ(t)�γΔ(t)) converges pointwise to (Σt�βt�λt�αt�γt) as Δ ↓ 0 for all t ≥ 0. The market price PΔ(t), the insider cumulative trad-ing process XΔ(t), and the market trading process ZΔ(t) converge weakly to Pt ,


Xt , and Zt , respectively, solutions of the system of stochastic differential equations(SDEs)

dZt = dXt + dYt� dPt = λt dZt�dXt =

{βt(Vt − Pt)dt� if t < T ,σy dB

yt + σy dBvt � if t ≥ T ,

with border conditions Z0 = X0 = Y0 = 0 and P0 = E[V0]. The insider expectedpayoff converges to Πt = αt(Vt −pt)2 + γt .

As in Kyle’s (1985) model, we could be tempted to argue that the limitingprofile (Σt�βt�λt�αt� γt) is an equilibrium of a continuous-time model in whichtrades and prices change continuously. In Section 5, however, we will show thatthis (continuity) property does not hold in our model. Hence, we can only in-terpret the continuous-time profile (Σt�βt�λt�αt� γt) as an asymptotically goodapproximation of the discrete-time equilibrium of Theorem 1 when agentstrade frequently. With this interpretation, we will refer to (Σt�βt�λt�αt� γt)as the limit equilibrium.

Theorem 2 reveals a number of important features of the limit equilibrium.A remarkable property is the existence of a finite time T , endogenously deter-mined, such that Σt = 0 for t ≥ T . That is, for Δ sufficiently small, the insideressentially reveals all her private information by time T . After T , the price al-ways matches the fundamental value of the asset. Despite this market efficiencythe insider is still able to collect positive rents (Π(t�0)= γT > 0) in t ∈ [T�∞).The source of these rents is the continuous inflow of new information that theinsiders gets by privately observing the evolution of the fundamental value. In-deed, one can show (see Theorem 3 below) that in the absence of these rents,either because Vt is constant or because the insider looses her capacity to trackVt , the insider would have no incentive to speed up her trading and marketefficiency would only be reached asymptotically (T = ∞).

To get a sense of how likely it is that market efficiency is reached in thelimit equilibrium, let us compare T and the average time 1/μ at which theannouncement date occurs. From the definition of T in Theorem 2, we canshow that

T ≤ 1μ

if Σ0 ≤(e2 − 3

2

)σ2v

μ∼ 2

σ2v

μ

Roughly speaking, the previous inequalities suggest that, on average, marketefficiency is reached when the insider’s initial (lumpy) private information Σ0

is less than twice her average cumulative inflow of new private informationσ2v /μ. Furthermore, one can show that as σv → ∞, the switching time T con-

verges to 0 and market efficiency is reach instantaneously. On the other hand,as σv ↓ 0, the switching time T diverges to +∞ and efficiency is only reached


asymptotically. The volatility coefficient σv determines the amount of infor-mation asymmetry. The following proposition shows that the higher is σv, thefaster the insider reveals her information, but also the larger is her profit. LetE[Πt] be the market’s best estimate of the insider’s expected continuation pay-off from time t on, that is, E[Πt] = αtΣt + γt . Because of the deterministicevolution of Σt , αt , and γt , E[Πt] is also the insider’s ex ante (at time 0, beforeobserving any signals) expected payoff-to-go from t onward.

PROPOSITION 2: In the limit equilibrium, the value of Σt weakly decreases withσv for all t ≥ 0. On the other hand, E[Πt] is equal to

E[Πt] = σvσy

μcosh(μ(T − t)+)�

which is increasing in σv for all t ≥ 0.

The more volatile is the fundamental value, the faster the price adjusts tothe current intrinsic value. However, this efficiency comes at a cost. Indeed,the insider is willing to trade away her private information faster because themarket maker compensates her for doing so. Hence, we expect market pricesto be more informative when the volatility of the fundamental value is higher.For example, when there is no volatility (σv = 0), market efficiency (Σt = 0)is reached only asymptotically as t → ∞ and the insider’s ex ante payoff isminimized.

In a discrete-time equilibrium, the market maker’s expected payoff is 0. Thisproperty is preserved in the limit equilibrium of Theorem 2. Thus, the liquiditytraders’ expected loss must equal the insider’s expected profit, E[Πt], whichaccording to Proposition 2 decreases monotonically with time in [0�T ) andstays constant after T . Thus, liquidity traders who place their orders late in thegame expect to make smaller losses.

Theorem 2 also shows that the market maker fulfills his obligation in a ratherstrong sense after T . He is concerned with setting prices so that pt = E[Vt|FM

t ].Theorem 2 implies that pt converges uniformly on compact sets to Vt in[T�∞).5 As a result, after T , the insider trading volumeXt behaves as a Brown-ian motion and has unbounded variation. It is also interesting to note thatXt −XT is independent of σv.

Finally, we note that the limit equilibrium satisfies the smooth-pasting con-dition

limt↑TΣt = 0

This is in contrast to the equilibria obtained in models that assume a fixed an-nouncement date (e.g., Kyle (1985)), where Σt does not approach 0 smoothly.

5This follows from the Skorohod representation theorem and the fact that Mt = Vt − pt con-verges weakly to (the continuous process) 0 for t ≥ T .


5. CONTINUOUS-TIME EQUILIBRIUM

In this section, we formulate the continuous-time counterpart of thediscrete-time model of Section 3 and show that the limit equilibrium of The-orem 2 is not an equilibrium of this model. However, we also show that in amodified continuous-time model where the insider’s flow of new informationis bounded, the limit equilibrium is an equilibrium.

Similar to the discrete-time model, we denote by Vt the fundamental value ofthe asset at time t which evolves as an arithmetic Brownian motion, Vt = σvBvt .

A strategy profile is a pair of processes (X�P), where Xt ∈ F It is the in-

sider’s cumulative trading volume up to time t, and Pt ∈ FMt is the price set

by the market maker at time t. Following the formulation of the continuous-time model in Back (1992), we restrict the trading process X to the class S ofcontinuous, F I

t -adapted square-integrable semimartingales. This is a technicalrequirement that allows us to write the insider payoff as a stochastic integralof the market price with respect to her trading strategy. More precisely, for agiven profile (X�P), the insider’s expected discounted payoff, E[Π(P�X)], isdefined as

E[Π(P�X)] = E

[VτXτ −

∫ τ

0Pt dXt − [X�P]τ

]�

where [X�P]t is the quadratic covariation between Xt and Pt .6 A continuous-time equilibrium is a profile (X�P) with the following properties: (i) given P ,X ∈ S maximizes E[Π(X�P)], and (ii) the price process P satisfies the equi-librium condition

Pt = E[Vt |FMt �X]� 0 ≤ t < τ

For the analysis that follows, we find it convenient to rewrite the insider’spayoff using the following identity

VτXτ =∫ τ

0Vt dXt +

∫ τ

0Xt dVt +

∫ τ

0d[X�V ]t �

where [X�V ]t is the quadratic covariation between Xt and Vt . Plugging thisidentity back into Π, taking expectation, and canceling the stochastic integralwith respect to the martingale Vt , we get

E[Π(P�X)] = E

[∫ ∞

0e−μt(Vt − Pt)dXt

+∫ ∞

0e−μt d[X�V ]t −

∫ ∞

0e−μt d[X�P]t

]�

6Intuitively, this term arises because the price paid by the insider is computed at the end ofthe period, and, therefore, it includes the effect of the insider’s last trade dXt . For a formalderivation, see equation (11) in Back (1992).


since τ is exponentially distributed with rate μ and is independent of F It .

Now we show that the strategy profile (P�X) associated to the limit equi-librium of Theorem 2 cannot be an equilibrium of the continuous-time model.Consider the insider’s expected payoff-to-go from time T onward,

ΠT(P�X)= E

[∫ ∞

T

e−μ(t−T)(Vt − Pt)dXt

+∫ ∞

T

e−μ(t−T) d[X�V ]t −∫ ∞

T

e−μ(t−T) d[X�P]t]

After time T , the market maker’s pricing strategy P is given by dPt = λT dZt ,where λT = σv/σy , and the insider’s cumulative volume of trade is a martingaleprocess such that dXt = σy[dBvt − dB

yt ]. Thus, Vt − Pt ≡ 0, the first stochas-

tic integral with respect to Xt has 0 expectation, and the quadratic covaria-tions between Xt and Vt and between Xt and Pt satisfy d[X�V ]t = σyσv dt andd[X�P]t = λtσ2

y dt = σyσv dt, respectively. It follows thatΠT(P�X)= 0 and soX cannot be a best reply to P . This shows that there is a discontinuity in theinsider’s payoff function as we move from discrete time to continuous time.Indeed, recall that the insider’s payoff ΠΔ of the discrete-time equilibrium ofTheorem 2 satisfies limΔ↓0Π

Δt = (σyσv)/μ > 0 for all t ≥ T . This discontinu-

ity is the result of the divergence of the insider’s trading rate βΔt to infinity fort ≥ T as Δ ↓ 0.7 In turn, this divergence is due to the existence of an unboundedflow of future private information. When the inflow of new information is small(for example, when σv = 0 because Vt is constant or the insider cannot track Vtafter t = 0), the insider would collect small rents after the market reaches fullefficiency. Therefore, the insider instead spends her private information slowlyand market efficiency is reached only asymptotically (T = ∞). In this case thelimit equilibrium of Theorem 2 is effectively an equilibrium of the continuous-time game. This and other properties of the equilibrium are summarized in thefollowing theorem.

We now assume that the insider’s strategy belongs to the space B of traderates β such that

E

[∫ ∞

0e−μt |βt |M2

t dt

]<∞(9)

Condition (9) rules out some bluffing schemes where the insider trades in the“wrong” direction and accumulates unbounded losses before accumulating un-bounded gains.

7In a modified model with quadratic transactional costs, the insider’s trading strategy would bebounded (as a referee suggested) and we expect the limit of discrete-time equilibria to be itselfan equilibrium of the continuous-time model.


THEOREM 3: Suppose the asset’s volatility σv(t) is a function of time, and letΓt be the insider’s cumulative inflow of private information from time t onward,that is,

Γt =∫ ∞

t

σ2v (t)dt

Assume that Γ0 <∞ and (Σ0 + Γ0)e−2μt > Γt for all t. When the insider’s strat-

egy space is constrained by (9), there exists a continuous-time linear Markovianequilibrium that satisfies

Σt = (Σ0 + Γ0)e−2μt − Γt� λt =

√2μ(Σ0 + Γ0)

σ2y

e−μt� βt =σ2y λt

Σt�(10)

αt = eμt

2

√σ2y

2μ(Σ0 + Γ0)� γt = αtΓt +

σ2y λt

4μ(11)

Under the conditions of Theorem 3, in equilibrium Σt ↓ 0 as t → ∞, butΣt > 0 for all t ≥ 0. More importantly, the trading rate βt remains bounded forall t ≥ 0, so the insider’s strategy is a process of bounded variation. When theflow of new information is substantial, the insider is happy to trade intenselyto exploit current arbitrage opportunities. Even though in the process she “in-forms” the market about what she knows now, new arbitrage opportunitieswill develop soon. In the limit equilibrium, she transfers all her information(initial + flow) by time T , but when this flow is relatively low, she is not willingto trade that fast.

6. CONCLUSIONS

The paper introduces a model that combines a random announcement timewith an insider who receives a flow of information. The new model producesa (limit) equilibrium with novel features. Two distinct regimes emerge. Beforethe endogenous time T , the insider is indifferent about how to consume herinformation stock, which includes the initial signal and the flow informationshe receives in the interval (0�T ]. Nevertheless, in equilibrium she exhausts allthis stock by time T , so that the market reaches full efficiency at time T . AfterT , she is eager to exhaust any additional piece of information immediately. Asshe does, she keeps the market fully informed until the public announcement,which reveals no further information.

The flow of new information, that in principle exacerbates the informationalasymmetry, in equilibrium induces the insider to release her information faster.Interestingly, the market is uniformly better informed and reaches full effi-ciency earlier when this source of informational asymmetry (the variation of


the innovation process) is larger. However, the larger the asymmetry, the largerare the rents extracted by the insider.

The analysis also exposes a potential difficulty with continuous-time mod-els. The natural discrete-time model has an equilibrium that, albeit difficultto construct explicitly, has a well defined limit as the period length decreasesto 0. However, this limit equilibrium is not an equilibrium of the correspondingcontinuous-time model.

APPENDIX

DEFINITION OF Ψ : To characterize the function Ψ(z), we find it convenientto introduce the change of variables

An = Σn

Σv� Bn = βnΣn√

ΣyΣv

Then equation (3) implies that (An+1�Bn+1)= F(An�Bn), where

FA(An�Bn)= 1 + A2n

An +B2n

� FB(An�Bn)= ρ[A2nBn

A2n −B4

n

]

Let

G1(A)=√

A

A− 1� G2(A)= √

A[1 − ρ]1/4� G3(A)= √A

Since in equilibrium βn must be positive for all n, a point (A�B) is feasi-ble only if FB(A�B) ≥ 0, that is, only if B ≤ G3(A). The function G1 is de-fined so that FA(A�G1(A)) = A. If B > G1(A), then FA(A�B) < A, and ifB < G1(A), then FA(A�B) > A. Similarly, the function G2 is defined so thatFB(A�G2(A))= B. If B >G2(A), then FB(A�B) > B, and if B <G1(A), thenFB(A�B) < B. As Figure 1 shows, the graphs of these functions partition the(A�B) space into five regions. In R1, F(A�B) is always to the left and higherthan (A�B), and any sequence {(An�Bn)} with initial point (A0�B0) in this re-gion eventually crosses the graph ofG3 and becomes infeasible. InR2, F(A�B)is always to the left and lower than (A�B). In R3, F(A�B) is always to the rightand lower than (A�B). In R4, F(A�B) is always to the right and higher than(A�B). R5 is the region of infeasible points. In Figure 1 we have also plottedfour sequences, each starting in a different region. A sequence that remainsfeasible must start in R2, R3, or R4, and any sequence that starts in R3 alwaysremain feasible, but not all sequences that start in R2 or R4 remain feasible.Sequences that start in R1 always become infeasible.

By definition, the intersection of the graphs ofG1 andG2 defines a stationarypoint (A� B) such that (A� B) = F(A� B) (see equation (7) for details of thisstationary point in terms of the original state variables).


FIGURE 1.—Partition induced by the functions G1, G2, and G3.

Now, by continuity of the vector field F , there exists a curve C , contained inR2 ∪ R4 and passing through (A� B), such that F(A�B) ∈ C for all (A�B) ∈C . That is, C is the largest subset of R

2 such that F(C) ⊂ C and (A� B) ∈ C .We do not have an analytical representation for C , but we can approximateit numerically. This curve is strictly increasing and it approaches the origin tothe left (but it does not contain it). Therefore, there exists a strictly increasingfunction ψ : (0�∞)→ (0�∞), such that (A�B) ∈ C if and only if B = ψ(A).For any initial A0 > 0, let B0 = ψ(A0). Then the sequence {(An�Bn)}, where(An+1�Bn+1) = F(An�Bn) for each n, is contained in C (that is, Bn = ψ(An)for all n ≥ 0) and, therefore, remains feasible forever. Moreover, (An�Bn)→(A� B) as n→ ∞. When A0 < A (respectively, A0 > A), B0 < B (B0 > B) and{(An�Bn)} is monotonically increasing (decreasing). In summary, for any givenΣ0 > 0, if we initialize

β0 =Ψ(Σ0)� where Ψ(Σ0)=√ΣyΣv

Σ0ψ

(Σ0

Σv

)�


we obtain a feasible sequence {(Σn�βn�λn�αn)}. Moreover, this sequence con-verges. In particular, {λn} is decreasing and converges to λ > 0. Therefore,there exists M > 0 such that λn ≤M for all n and

∞∑n=1

ρn

λn<+∞

LEMMA 1: Assume that the market maker’s strategy {Pn} is specified by a deter-ministic sequence {λn} ⊂ R++. Let

S =∞∑n=1

ρn

λn

If S = ∞, then Πn(p�V )= ∞ for all n≥ 0 and (p�V ) ∈ R2. If S <∞ and there

isM > 0 such that λn <M and ρλn/λn+1 ≤ 1 for all n≥ 0, then there exist positivesequences {αn} and {γn} such that λnαn+1 ≤ 1/2 and ρΠn(p�V )= αn(p− V )2 +γn for all n≥ 0.

PROOF: For each n≥ 0 and each k≥ 0, consider the finite horizon problemfor the insider where the fundamental value is made public at the end of periodn+k if it has not been publicly revealed before. LetΠk�n(p�V ) be the insider’soptimal discounted value from period n onward in this problem, when the priceand fundamental value in period n − 1 are (p�V ). Obviously, Πk�n(p�V ) ≤Πn(p�V ) (because the insider can always choose xs = 0 for all s > n+ k) andlimk→∞Πk�n(p�V )=Πn(p�V ) for all n≥ 0 and all (p�V ) ∈ R

2.We first show inductively in k that for each n, either

Πk�n(p�V )= ak�n

λn(V −p)2 + bk�n

λnΣv + ck�nλnΣy(12)

for some constants (ak�n� bk�n� ck�n) or Πk�n ≡ ∞. When k= 0,

Π0�n(p�V )= max(V −p− λnx)x

= (V −p)2

4λn�

so Π0�n satisfies (12) with a0�n = 1/4 and b0�n = c0�n = 0 for all n≥ 0. By induc-tion, assume first thatΠk�n+1 satisfies (12) for a given (k�n). We then show thateither Πk+1�n also satisfies (12) or Πk+1�n ≡ ∞. We have that

Πk+1�n(p�V )= maxx∈R

(V −p− λnx)x+ ρE

[Πk�n+1(V +Wn�p+ λn(x+Yn))

]


= maxx∈R

(V −p− λnx)x

+ ρ[ak�n+1

λn+1[(V −p− λnx)2 +Σv + λ2

nΣy]

+ bk�n+1

λn+1Σv + ck�n+1λn+1Σy

]

When ρak�n+1λn/λn+1 < 1, the quadratic objective function is concave andΠk+1�n satisfies (12) with

ak+1�n = 14

[1 − ak�n+1

ρλn

λn+1

]−1

�(13)

bk+1�n = ρλn

λn+1[ak�n+1 + bk�n+1]� ck+1�n = ρ

[ak�n+1

λn

λn+1+ ck�n+1

λn+1

λn

](14)

When ρak�n+1λn/λn+1 ≥ 1, the quadratic objective function is convex andΠk+1�n ≡ ∞. By induction, now assume instead that Πk�n+1 ≡ ∞. ThenΠk+n+1−s�s ≡ ∞ for all s= 0� � n. This concludes the proof by induction.

Let us now assume that∑ρn/λn = ∞. In this case, we will show that

Πk�n(p�v)→ ∞ as k→ ∞, for all n and (p�v).Since a0�n = 1/4 and ρλn/λn+1 > 0 for all n ≥ 0, it is easy to see (by in-

duction) that (13) implies ak�n > 1/4 for all k ≥ 1 and n ≥ 0. Fix n ≥ 0. Forany k ≥ 1, if there exists j ∈ {1� �k} such that 1 ≤ ak−j�t+jρλn+j/λn+j+1, thenΠk−j+1�n+j−1 ≡ ∞, which implies that Πk�n ≡ ∞ and Πn ≡ ∞. Conversely, if1> ak−j�t+jρλn+j/λn+j+1 for all j ∈ {1� �k}, then (14) implies that

bk�n ≥ ρλn

λn+1

[14

+ bk−1�n+1

]≥ ρλn

λn+1

[14

+ ρλn+1

λn+2

[14

+ bk−2�n+2

]]≥ · · ·

≥ λn

4

[ρ

λn+1+ · · · + ρk

λn+k

]

Note that∞∑j=1

ρj

λn+j= 1ρn

∞∑j=t+1

ρj

λj= 1ρn

[ ∞∑j=1

ρj

λj−

n∑j=1

ρj

λj

]= ∞

Therefore,

Πn(p�v)≥Πk�n(p�v)≥ Σv

λnbk�n ≥ Σv

4

k∑j=1

ρj

λn+jfor all k≥ 1�

which implies again thatΠn ≡ ∞ since the last term converges to ∞ as k→ ∞.Thus, either way, Πn ≡ ∞.


Finally, assume that∑ρn/λn < ∞ and ρλn/λn+1 ≤ 1 for all n ≥ 1. In this

case, we show that each Πn(p�V ) is a quadratic function of (V −p).Since a0�n = 1/4, it is easy to show by induction that 1/4 < ak�n < 1/2

for all k ≥ 1 and n ≥ 0. The function f (a�d) = [4(1 − ad)]−1 is increasingin a and d when ad < 1. Since a1�n+1 > 1/4 = a0�n+1 for all n ≥ 0, a2�n =f (a1�n+1� dn) > f(a0�n+1� dn) = a1�n for all n ≥ 0. Repeating this argument for-ward, we conclude that {ak�n}∞

k=1 is an increasing sequence and it must con-verge. Let αn = limk→∞ ak�n/λn. Now ak�n+1 < 1/2 for all k and ρλn/λn+1 ≤ 1imply that λnαn+1 ≤ 1/2.

Again, ak�n < 1/2 for all k≥ 0 and n≥ 0 imply that

bk�n ≤ ρλn

λn+1

[12

+ bk−1�n+1

]≤ · · · ≤ λn

2

[ρ

λn+1+ · · · + ρk

λn+k

]

<λn

2ρn

∞∑j=t+1

ρj

λj<∞

By induction in k, we now show that bk�n < bk+1�n for all k ≥ 0 and n ≥ 1.Clearly b0�n = 0 < b1�n for all n ≥ 0. Since ak�n+1 < ak+1�n+1, if the inequalityholds for (k�n), then

bk+1�n = dn[ak�n+1 + bk�n+1]< dn[ak+1�n+1 + bk+1�n+1] = bk+2�n

That is, for each n ≥ 0, the sequence {bk�n}∞k=0 is increasing and hence it must

converge. Solving (14), we obtain

ck�n = 1λn

k∑j=1

ρjλ2n+j−1

λn+jak−j�n+j

One can show that for each n ≥ 0, the sequence {ck�n}∞k=1 is increasing, and

since λs ≤M and aj�s < 1/2 for all j and s,

ck�n ≤ M2

2λn

k∑j=1

ρj

λn+j<

M2

2λnρn

∞∑j=t+1

ρj

λj<∞

and the sequence must converge. Let γn = limk→∞[bk�nΣv/λn+ ck�nλnΣy]. ThenΠn(p�v)= αn(v−p)2 + γn. Q.E.D.

LEMMA 2: Let {βn} be an arbitrary deterministic strategy for the insider. As-sume that {λn} satisfies the equilibrium condition pn = E[V |{βn}�FM

n ]. Then theinsider’s expected payoff when she follows {βn} is finite.


PROOF: The insider’s payoff satisfies

Π =η∑n=0

(Vη −pn)xn =η∑n=0

βn(Vη −pn)(Vn −pn−1)

Let us write pn and Vn in terms of the primitive stochastic sequences {vn} and{yn} with v0 = V0. We have that Vn =∑n

n=0 vn. In addition,

pn = pn−1 + λn(βn(Vn −pn−1)+ yn)

= (1 − λnβn)pn−1 + λnβnn∑n=0

vn + λnyn

Suppose that there exist sequences A(n), B(k�n), and C(k�n) such thatA(0)= 1, B(0�0)= C(0�0)= B(k�n)= C(k�n)= 0 for k> n, and

pn =A(n)p−1 +n∑k=0

[B(k�n)vk +C(k�n)yk]

It follows that

A(n)= (1 − λnβn)A(n− 1)�

B(k�n)= λnβn + (1 − λnβn)B(k�n− 1) for 0 ≤ k≤ n�C(k�n)= (1 − λnβn)C(k�n− 1) for 0 ≤ k< n and

C(n�n)= λn

For j ≥ i, let ψ(i� j) :=∏j

k=i(1 − λkβk). Iterating the recursions above, we get

A(n)=ψ(0� n)� B(k�n)=n∑j=kψ(j + 1� n)λjβj�

C(k�n)=ψ(k+ 1� n)λk

The insider’s conditional expected payoff given η is

E[Π|η] =η∑n=0

βnE[(Vη −pn)(Vn −pn−1)|η]

=η∑n=0

βnE[(Vn −pn)(Vn −pn−1)] =η∑n=0

[D1n +D2

n +D3n]�


where

D1n = βnA(n)A(n− 1)p−1�

D2n = βn

n∑k=0

(1 −B(k�n− 1))(1 −B(k�n))Σv(n)�

D3n = βn

n∑k=0

C(k�n− 1)C(k�n)Σy�

Σv(0) = Σ0, and Σv(n) = Σv for n > 0. We now bound these terms. We havethat

λn = βnΣn

β2nΣn +Σy

⇒ 1 − λnβn = Σy

β2nΣn +Σy ∈ (0�1) for all n≥ 0

Hence, A(n) is a nonnegative decreasing sequence with 0 ≤A(n)≤A(0)≤ 1.Moreover, 0 ≤ B(k�n)≤ B(0� n) for all 0 ≤ k≤ n. The sequence B(0� n) satis-fies the recursion

B(0� n)= λnβn + (1 − λnβn)B(0� n− 1)� B(0�0)= 0

Therefore, 0 ≤ B(k�n) ≤ B(0� n) ≤ 1. Also, the function f (x) = x/(ax2 + 1)reaches its global maximum at x= ±√1/a. Therefore,

|λn| ≤√Σn

1 +Σy ≤ 12

√Σn

Σy� |βn|(1 − λnβn)= λnΣy

Σn≤ 1

2

√Σy

Σn

Thus we obtain the bounds

|D1n| = |βn|(1 − λnβn)

n−1∏k=0

(1 − λkβk)2 ≤ 12

√Σy

Σn≤ 1

2

√Σy

Σv�

since 1 − λkβk ∈ (0�1) and Σn ≥ Σv for all n≥ 1. For the second term we havethat

|D2n| = |βn|(1 − λnβn)

n∑k=0

(1 −B(k�n− 1))2Σv(n)

≤ n+ 12

√Σy

Σvmax{Σ0�Σv}


Finally, for the third term we have that

|D3n| = |βn|(1 − λnβn)

n∑k=0

C(k�n− 1)2Σy ≤ Σy

2

√Σy

Σv

n∑k=0

λ2k�

where the last inequality uses the fact that C(k�n− 1)≤ λk. The recursion forΣn+1 implies that Σn+1 ≤ Σv +Σn. Therefore, Σn ≤ nΣv +Σ0. We conclude that

|D3n| ≤

Σy

2

√Σy

Σv

n∑k=0

Σn

4Σy≤ 1

8

√Σy

Σv

(n+ 1)(n+ 2)2

max{Σ0�Σv}

Combining all the pieces together we get that

|Dn| ≤ 12

√Σy

Σv

[1 +

(1 + n+ 2

8

)(n+ 1)max{Σ0�Σy}

]≤Kn2

for some constant K. Therefore,

E[Π] = E[E[Π|η]] =∞∑n=0

ρnDn <∞ Q.E.D.

LEMMA 3: Choose β0 < Ψ(Σ0), and let {λn} and {βn} be the correspondingstrategies generated by (3) and (4) for the market maker and the insider. Then∑ρn/λn = ∞ and the insider can make infinite profits. Moreover, {βn} is not a

best reply against {λn}.PROOF: We show that if β0 < Ψ(Σ0), then

∑ρn/λn = ∞. Lemma 1 above

then implies that the insider’s expected payoff is unbounded. However, byLemma 2, {βn} generates finite profits. Therefore, {βn} is not optimal against{λn}.

When β0 <Ψ(Σ0), the sequence {(An�Bn)} lies below C and remains feasi-ble forever. Moreover, for some finite N , (An�Bn) ∈ R3 for all n ≥N . There-fore, An <An+1 for all n≥N and An → ∞. Recall that the graphs of G1 andG2 intersect at (A� B), and that (A�B) ∈R3 andA≥ A imply that B≤G1(A).The function h(A) = (A − 1)2/[A(A − 2)] is decreasing for all A > 2, andh(A)→ 1 as A→ ∞. Let ω ∈ (ρ�1). Without loss of generality, assume thatN is such that AN ≥ A and h(AN) ≤ ω/ρ. Then Bn ≤ G1(An) for all n ≥ Nand, therefore, for all n≥N ,

Bn+1 = FB(An�Bn)= ρ[A2nBn

A2n −B4

n

](15)

≤ ρ[

A2nBn

A2n − [G1(An)]4

]= ρh(An)Bn ≤ωBn


Since BN ≤ B, this implies that Bn ≤ Bωn−N for all n≥N . From equation (3),

λn = βnΣn

β2Σn +Σy = AnBn

An +B2n

√Σv

Σy< Bn

√Σv

Σy

Since we would like to show that∑ρn/λn = ∞, we need a tighter upper bound

on Bn. Note, however, that

Bn+1 = FB(An�Bn)= ρ[A2nBn

A2n −B4

n

]≥ ρBn for all n≥ 0�

so there is not a lot of slack in the previous upper bound (15) for Bn+1.For any ε > 0, let N∗ >N be such that BωN∗−N < ε. Then, for all n≥N∗,

An+1 = FA(An�Bn)= 1 + A2n

An +B2n

≥ 1 + An

1 + ε2/An

≥ 1 +An

[1 − ε2

An

]=An + (1 − ε2)

Let e= 1−ε2. ThenAN∗+n ≥AN∗ +ne > ne for all n≥N∗. Feeding this boundback into (15), we obtain that

BN∗+n+3 ≤ ρh((n+ 2)e)BN∗+2+n ≤ · · ·≤ ρnh((n+ 2)e)h((n+ 1)e) · · ·h(3e)BN∗+3

Choose ε < 1/4 so that ε2 < 1/16. Then, for all k≥ 3,

h(ke)= [k− 1 − kε2]2

[k− kε2][k− 2 − kε2]= 1 + 1

k(k− 2)− 2k(k− 1)ε2 + k2ε4

< 1 + 1k[k− 2 − 2(k− 1)ε2] < 1 + 8

k[7k− 15] ≤ 1 + 4k2

Let

Hn =[

1 + 412

][1 + 4

22

]· · ·

[1 + 4

n2

]�

an = 1Hn

=[

12

12 + 4

]· · ·

[n2

n2 + 4

]


Note that [1 + 4/12][1 + 4/22] = 10. Hence, BN∗+n+3 < ρnBN∗+3Hn+2/10. There-

fore, √Σv

Σy

∑n≥1

ρn

λn>∑n≥1

ρn

Bn>∑n≥1

10ρN∗+3+n

ρnHn+2BN∗+3= 10BN∗+3

ρN∗+3

∑n≥3

an

Gauss’s test (see, for example, Knopp (1990)) states that if

an+1

an= 1 − c

n− gn

nε�

where ε > 1 and {gn} is bounded, then∑an converges when c > 1 and diverges

when c ≤ 1. In our case,

an+1

an= (n+ 1)2

(n+ 1)2 + 4= 1 −

[4n2

(n+ 1)2 + 4

]1n2�

so c = 0 and ε = 2. Therefore∑an = ∞, so

∑ρn/λn = ∞ and the insider

makes infinite profits. Q.E.D.

The following lemma is used in the proof of Theorem 2.

LEMMA 4: Let {fn} be a sequence of convex functions on [T�∞) (where T ∈R is arbitrary). Assume that fn converges pointwise to 0. That is, for all t ≥ T ,fn(t)→ 0 as n→ ∞. Then, for each t > T , ∂fn(t)→ {0}.

PROOF: By contradiction, assume that there exists t∗ > T and a subgradientsn ∈ ∂fn(t∗) for each n, such that {sn} does not converge to 0. Without loss ofgenerality, assume that sn → s < 0. Then, for each t ∈ [T� t∗), fn(t) ≥ fn(t∗)+sn(t − t∗), and taking limits as n→ ∞, we obtain 0 ≥ −s(t∗ − t) > 0, which is acontradiction. Q.E.D.

PROOF OF THEOREM 2: To emphasize the dependence of the discrete-timeequilibrium on the length of a period, let us denote by {(ΣΔn �βΔn �λΔn �αΔn �γΔn )}the discrete-time equilibrium of Theorem 1 and denote by (ΣΔ� βΔ� λΔ� αΔ� γΔ)the corresponding stationary equilibrium for an arbitrary Δ > 0. Since Σ0 > 0,it follows that Σ0 > Σ

Δ ∼ O(√Δ) for Δ sufficiently small. As a result, the

sequences {ΣΔn } and {λΔn } are monotonically decreasing, while the sequences{βΔn } and {αΔn } are monotonically increasing in n for Δ sufficiently small.Also, recall that the profile (ΣΔ(t)�βΔ(t)�λΔ(t)�αΔ(t)�γΔ(t)) is a continuous-time piecewise-linear approximation of the discrete-time equilibrium suchthat (ΣΔ(t)�βΔ(t)�λΔ(t)�αΔ(t)�γΔ(t)) = (ΣΔn �β

Δn/Δ�λ

Δn �α

Δn �γ

Δn ) for all t ∈

[nΔ� (n+ 1)Δ).


The remaining proof is divided into two parts. In Part I, we show the point-wise convergence of (ΣΔ(t)�βΔ(t)�λΔ(t)�αΔ(t)�γΔ(t)) to (Σt�βt�λt�αt� γt) asΔ ↓ 0. In Part II, we prove the weak convergence of (PΔ(t)�XΔ(t)�ZΔ(t)) to(Pt�Xt�Zt).

Part I. The proof of this part is organized as follows. First, we show thatequations (3) and (4), which characterize the evolution of {(ΣΔn �βΔn �λΔn �αΔn �γΔn )}, converge to a system of ordinary differential equations as Δ ↓ 0. Thenwe show that the solution of these ordinary differential equations (ODEs)defines a continuous-time profile (Σ(t)�β(t)�λ(t)�α(t)�γ(t)) that is arbitrar-ily closed (as Δ ↓ 0) to (ΣΔ(t)�βΔ(t)�λΔ(t)�αΔ(t)�γΔ(t)) for all t < T , whereT = sup{t > 0 :Σ(s) > 0�∀s < t}. As in the discrete-time case, β(0) is a free pa-rameter for this continuous-time profile. Finally, we show that β(0) is uniquelydetermined using two properties of the discrete-time equilibrium: (i) ΣΔn is de-creasing in n, which provides a lower bound on β(0), and (ii) βΔnΣ

Δn is decreas-

ing in n, which provides an upper bound on β(0). We conclude the first part ofthe proof, showing that these upper and lower bounds coincide.

For notational convenience, let us define qΔ(t) := βΔ(t)ΣΔ(t) and rΔ(t) :=Δ/ΣΔ(t). The recursive equations (3) and (4) imply that

ΣΔ(t +Δ)−ΣΔ(t)Δ

= σ2v − (qΔ(t))2

σ2y + (qΔ(t))2rΔ(t)

�(16)

qΔ(t +Δ)− qΔ(t)Δ

=[σ4y (e

−μΔ − 1)/Δ+ (qΔ(t))4rΔ(t)/ΣΔ(t)

σ4y − (qΔ(t))4(rΔ(t))2

]qΔ(t)�

λΔ(t)= qΔ(t)

σ2y + (qΔ(t))2rΔ(t)

� αΔ(t)= 1 − λΔ(t)βΔ(t)Δ2λΔ(t)

�

γΔ(t +Δ)− γΔ(t)Δ

= eμΔ − 1Δ

γΔ(t)− 1 − 2λΔ(t)βΔ(t)Δ2λΔ(t)(1 − λΔ(t)βΔ(t)Δ)

(σ2v + (λΔ(t))2σ2

y

)eμΔ

For a given t, suppose that lim supΔ↓0(βΔ(t))2ΣΔ(t) <∞. Then (qΔ(t))2rΔ(t)/√

Δ is negligible for Δ sufficiently small and, as Δ ↓ 0, the system of equa-tions (16) converges to

Σ(t)= σ2v − q(t)2

σ2y

� q(t)= −μq(t)�(17)

λ(t)= 12α(t)

= q(t)

σ2y

� γ(t)= μγ(t)− σ2v + λ(t)2σ2

y

2λ(t)(18)

Since Σ0 > 0, the condition lim supΔ↓0(qΔ(t))2rΔ(t)= 0 is satisfied at t = 0 (oth-

erwise lim infΔ↓0 λΔΔ = ∞). It follows (by continuity) that the convergence above


holds for all t ∈ [0�T ) for some positive T > 0. Then by integrating (17) in thisrange, we obtain a continuous-time profile (Σ(t)�β(t)�λ(t)�α(t)�γ(t)) givenby

Σ(t)= Σ0 + σ2v t −

(β(0)Σ0)2

2μσ2y

(1 − e−2μt)�(19)

q(t)= β(0)Σ0e−μt for t < T�

for some constant of integration β(0). Note that for this continuous-time so-lution, the condition lim supΔ↓0 (q

Δ(t))2rΔ(t) = 0 reduces to Σ(t) > 0. Hence,given β(0), T is uniquely determined as the smallest (positive) solution of theequation Σ(t)= 0. We denote by T(β(0)) this value which solves

0 = Σ0 + σ2v T − (β(0)Σ0)

2

2μσ2y

(1 − e−2μT )

Suppose β(0) is small enough so that T(β(0)) = ∞. Then Σ(t) > 0 for allt ≥ 0 and limΔ↓0(Σ

Δ(t)�βΔ(t)) = (Σ(t)�β(t)) for all t ≥ 0. But in this caselimt→∞Σ(t)= ∞, which implies that limt→∞ΣΔ(t)= ∞ for Δ sufficiently small,contradicting the monotonicity of ΣΔn . As a result, T(β(0)) <∞ and so β(0)is bounded below by βL(0) such that Σ(t)= Σ(t)= 0 at t = T(βL(0)). That is,βL(0) satisfies

σ2v − (βL(0)Σ0)

2

σ2y

e−2μT = 0

where T solves

0 = Σ0 + σ2v T − σ2

v

(e2μT − 1

2μ

)

Suppose now that β(0) > βL(0) so that T(β(0)) <∞. Then the fact that ΣΔnis monotonically decreasing in n for all Δ and equation (19) lead to

Σt = limΔ↓0ΣΔ(t)

=⎧⎨⎩Σ0 + σ2

v t −(β(0)Σ0)

2

2μσ2y

(1 − e−2μt)� if t < T(β(0)),

0� if t ≥ T(β(0)).In what follows, we show that for t ≥ T(β(0)),

limΔ↓0

ΣΔ(t +Δ)−ΣΔ(t)Δ

= 0(20)


Equation (16) and the fact that ΣΔ(t) and qΔ(t) are decreasing functions oft imply that ΣΔ(t) is convex for all t ≥ T(β(0)). Hence, by Lemma 4, ΣΔ(t)satisfies equation (20). We can use this result together with the first equationin (16) to show that for t ≥ T(β(0)),

limΔ↓0qΔ(t)= lim

Δ↓0βΔΣΔ = σvσy a.e.

This follows from the fact that qΔ(t) is decreasing in t for all Δ and, therefore,is bounded, which implies

limΔ↓0(qΔ(t))2rΔ(t)= lim

Δ↓0(βΔ(t)ΣΔ(t))2 Δ

ΣΔ(t)= 0

since ΣΔ(t)≥ ΣΔ ≥ σv(Δ+ √Δ/μ). On the other hand, from equation (19) we

get that

limt↑T(β0)

limΔ↓0qΔ(t)= β(0)Σ0e

−μT(β(0))

Hence, unless β(0)Σ0e−μT(β(0) = σvσy (or equivalently β(0)= βL(0)), the limit

function q(t) would have a discontinuity at t = T(β(0)). But from the secondequation in (16), such a discontinuity is not possible because the term[

σ4y (e

−μΔ − 1)/Δ+ (qΔ(t))4rΔ(t)/ΣΔ(t)

σ4y − (qΔ(t))4(rΔ(t))2

]qΔ(t)

is uniformly bounded in t and Δ8, which by the Arzelà–Ascoli theorem impliesthat the limit function q(t) is continuous.

Part II. To prove the weak convergence of (PΔ(t)�XΔ(t)�ZΔ(t)) to (Pt�Xt�Zt), we introduce the price gap processesMΔ

t := V Δt −PΔt andMt := Vt−Pt , and

show thatMΔt converges weakly toMt . Specifically, we will invoke Theorem 2.1

in Prokhorov (1956) and prove the convergence of the finite-dimensional dis-tributions of MΔ

t to those of Mt , and then show the compactness of MΔt in Δ.

For any Δ > 0, the corresponding discrete-time equilibrium characterizesthe values of MΔ

t , βΔt , and λΔt only at the discrete sequence of times {iΔ}i≥0.To extend these functions to R+, we introduce the following notation: for anyt > 0, we define, nΔt := lims↑ts/Δ and tΔ := nΔt Δ, and for any function f Δt , wedefine f Δt := f Δ

tΔ. Using a slight abuse of notation, we redefine the continuous

piecewise linear version of MΔt for any t > 0 as

MΔt =MΔ

t (1 − λΔt βΔt (t − tΔ))+ σv(Bvt −BvtΔ)− λΔt σy(Byt −By

tΔ)�

8This follows from the fact that in equilibrium, qΔ(t) is nonnegative and decreasing in t, and(qΔ(t))4(rΔ(t))2 ≤ σ4

y (1 − ρ) (see Figure 1).


with border condition MΔ0 = V0 − E[V0] Since we are only concerned with the

weak convergence of MΔt , we will simplify the notation, replacing the term

σv(Bvt − Bv

tΔ)− λΔt σy(Byt − By

tΔ) by σΔt (Bt − BtΔ), where Bt is a Wiener process

and (σΔt )2 = σ2

v + σ2y (λ

Δt )

2. Iterating the recursion for MΔt above, we get that

MΔt =MΔ

0 A(0� nΔt )+

nΔt∑k=0

A(k+ 1� nΔt )σΔkΔ

(B(k+1)Δ∧tΔ −BkΔ

)�

where

A(j�nΔt ) :=nΔt∏k=j

(1 − λΔkΔβΔkΔ

(min{(k+ 1)Δ� tΔ} − kΔ))

Since both λΔt and βΔt are deterministic processes, it follows that MΔt is a

Gaussian process. Hence, its finite-dimensional distribution is fully character-ized by its mean and variance–covariance processes. For t > 0, we have

μΔt := E[MΔt ] =MΔ

0 A(0� nΔt )

From Part I, we know that (i) λΔt converges pointwise to a smooth, strictly pos-itive, and bounded function λt for all t ≥ 0, and (ii) the function βΔt is nonde-creasing in t for all t ≥ 0 and converges pointwise to a smooth nondecreasingfunction βt in [0�T ) and to infinity in t ≥ T . We conclude that as Δ ↓ 0, then

limΔ↓0μΔt = 1(t < T)e− ∫ t

0 λ(s)β(s)ds�

where 1(t < T) is the indicator function equal to 1 if t < T and equal to 0otherwise. Similarly, if we define Γ Δ(t� t ′) := E[(MΔ

t − μΔt )(MΔt − μΔt′)] to be

the variance–covariance process of MΔt , then as Δ ↓ 0, we get (for t < t ′)

limΔ↓0Γ Δ(t� t ′)= 1(t ′ < T)e− ∫ t′

t λsβs ds

∫ t

0e−2

∫ ts λuβu du(σ2

v + λ2sσ

2y ) ds

Consider now the limiting gap process Mt = Vt −Pt . It follows from the systemof SDEs in Theorem 2 that Mt satisfies the SDE

dMt = −λtβt Mt + σy dByt − λtσy dByt for all t < T

and Mt = 0 for t ≥ T . As a result, Mt is also a Gaussian process. Furthermore,for t < T , we can integrate the SDE above to get

Mt =M0e− ∫ t

0 λsβs ds +∫ t

0e− ∫ t

s λuβu du(σv dBvs − λsσy dBys )


It is a matter of simple calculations to show that the mean process E[Mt] andthe variance–covariance process Γ (t� t ′)= E[(Mt − E[Mt])(Mt′ −E[Mt′ ])] co-incide with limΔ↓0μ

Δt and limΔ↓0 Γ

Δ(t� t ′) computed above. We conclude thatthe finite-dimensional distribution of MΔ

t converges to the finite-dimensionaldistribution of Mt for all t ≥ 0. In particular, it is worth noticing that MΔ

T con-verges weakly to 0 as Δ ↓ 0.

We now prove that {MΔt :Δ> 0} is tight in [0�T ] for an arbitrary T > 0. For

this we show that for every ε > 0

limδ↓0

lim supΔ↓0

P

(sup

|t−s|≤δ|MΔ

t −MΔs | ≥ ε

)= 0

For 0 ≤ s < t ≤ T such that iΔ≤ s ≤ (i+ 1)Δ and jΔ≤ t ≤ (j + 1)Δ, it followsthat

|MΔt −MΔ

s | ≤ |MΔjΔ −MΔ

iΔ| +∣∣MΔ

(i+1)Δ −MΔiΔ

∣∣+ ∣∣MΔ(j+1)Δ −MΔ

jΔ

∣∣For notational convenience, let use introduce the notation MΔ

i =MΔiΔ (a sim-

ilar notation is used for βΔi , λΔi , σΔi , ΣΔi , and Bi). Defining δΔ := δ/Δ + 1,TΔ := T/Δ, and T Δ := T /Δ, for Δ sufficiently small, the previous inequal-ity implies that

P

(sup

|t−s|≤δ|MΔ

t −MΔs | ≥ ε

)≤ 3P

(sup

|j−i|≤δΔ|MΔ

j −MΔi | ≥ ε/3

)

Using the recursion for MΔi , we get that

|MΔj −MΔ

i | ≤j−1∑k=iλΔkβ

ΔkΔ|MΔ

k | +∣∣∣∣∣j−1∑k=iσΔk (Bk+1 −Bk)

∣∣∣∣∣and so

P

(sup

|j−i|≤δΔ|MΔ

j −MΔi | ≥ ε

)

≤ P

(sup

|j−i|≤δΔ

j−1∑k=iλΔkβ

Δk |MΔ

k |Δ≥ ε/2)

+ P

(sup

|j−i|≤δΔ

∣∣∣∣∣j−1∑k=iσΔk (Bk+1 −Bk)

∣∣∣∣∣≥ ε/2)

From Part I we know that λΔ(t) is uniformly bounded in Δ and t and strictlypositive. So, for the purpose of the result that we need to prove, we can conve-


niently assume that without loss of generality (w.l.o.g.), λΔk = 1 (and σΔk = 1). Itfollows that for the last term on the right that

limδ↓0

lim supΔ↓0

P

(sup

|j−i|≤δΔ

∣∣∣∣∣j−1∑k=iσΔk (Bk+1 −Bk)

∣∣∣∣∣≥ ε/2)

= 0

(e.g., by invoking Lévy’s theorem on the modulus of continuity for Brownianmotion). Hence, to complete the proof, it is now sufficient to show that

limδ↓0

lim supΔ↓0

P

(sup

|j−i|≤δΔ

j−1∑k=iβΔk |MΔ

k |Δ≥ ε)

= 0(21)

From the definition of MΔk (and the assumption λΔk = σΔk = 1), we get that

βΔk |MΔk | ≤ |MΔ

0 |βΔkk−1∏j=0

(1 −βΔj Δ)

+∣∣∣∣∣k−1∑j=0

βΔk

(k−1∏n=j+1

(1 −βΔnΔ))(Bj+1 −Bj)

∣∣∣∣∣Suppose T < T . Then in the region t ∈ [0�T ], the function βΔt is uniformlybounded in Δ and t, and the condition in equation (21) will follow. So, let usassume that T ≥ T . From Part I, it follows that there exist positive constants

K1

ΣΔt≤ βΔt ≤ K2

ΣΔtfor t ∈ [0�T ]

This follows from the fact that βΔt ΣΔt converges to a positive bounded function

as Δ goes to zero. Furthermore, given the limiting behavior of ΣΔt as Δ ↓ 0, onecan show that for any t ∈ [T�T ], there exists a positive constant Kt (indepen-dent of Δ) such that ΣΔs ≤ ΣΔt +Kt(t− s)2 for all s ∈ [0� t]. As a result, in [T�T ]we get that

sup|j−i|≤δΔ

|MΔ0 |Δ

j−1∑k=iβΔk

k−1∏j=0

(1 −βΔj Δ)

≤ |MΔ0 |(δ+Δ) max

TΔ≤k≤T Δ

{βΔk exp

(−

k−1∑j=0

βΔj Δ

)}

≤ |MΔ0 |(δ+Δ) max

TΔ≤k≤T Δ

{K2

ΣΔkexp

(−∫ kΔ

0

K1 dt

ΣΔk +KkΔ(kΔ− s)2

)}


= |MΔ0 |(δ+Δ) max

TΔ≤k≤T Δ

{K2

ΣΔkexp

(− K1√

KkΔΣΔk

arctan(

kΔ√KkΔΣ

Δk

))}Δ↓0−→ 0�

where the convergence follows from the fact that ΣΔt → 0 as Δ ↓ 0 for anyt ≥ T . It follows from the previous derivation that there exists a constant Kindependent of Δ such that

maxTΔ≤k≤T Δ

{βΔk

k−1∏n=0

(1 −βΔnΔ)}

≤ K

Finally, we have that

P

(sup

|j−i|≤δΔ

j−1∑k=iΔ

∣∣∣∣∣k−1∑j=0

βΔk

(k−1∏n=j+1


∣∣∣∣∣≥ ε)

≤ P

((δ+Δ)

× maxTΔ≤k≤T Δ

{∣∣∣∣∣k−1∑j=0

βΔk

(k−1∏n=j+1


∣∣∣∣∣}

≥ ε)

≤ P

((δ+Δ)K max

TΔ≤k≤T Δ

{∣∣∣∣∣k−1∑j=0

j∏n=0

(1 −βΔnΔ)−1(Bj+1 −Bj)∣∣∣∣∣}

≥ ε)

≤ (δ+Δ)2K2

ε2E

[T Δ−1∑j=0

j∏n=0

(1 −βΔnΔ)−2Δ

]

≤ (δ+Δ)2K2 Tε2

max0≤j≤T Δ

j∏n=0

(1 −βΔnΔ)−2

The third inequality uses Doob’s inequality. From Part I we know that βΔn isan increasing function of n and that it converges to βΔ∞ = K/

√Δ for a fixed

constant K independent of Δ. As a result, max0≤j≤T Δ

∏j

n=0(1 − βΔnΔ)−2 is uni-

formly bounded. We conclude that the limδ↓0 lim supΔ↓0 of the probability aboveis equal to zero as required. Q.E.D.


PROOF OF PROPOSITION 2: Recall from Theorem 2 that Σt satisfies

Σt = Σ0 + σ2v t − σ2

v e2μT

[1 − e−2μt

2μ

]for t < T

and ΣT = 0 for all t ≥ T , where T ≥ 0 is the unique solution to

Σ0 + σ2v T = σ2

v

[e2μT − 1

2μ

]

Since T decreases with σv, it suffices to prove that Σt decreases with σv fort < T .

In what follows, and without lost of generality, we will normalize the valueof μ such that 2μ= 1 (this is equivalent to rescaling time). With this normal-ization, the derivative of Σt (t < T ) with respect to σ2

v is equal to

∂Σt

∂σ2v

= t − eT (1 − e−t)− σ2v e

T (1 − e−t)∂T

∂σ2v

for t < T

In addition, from the definition of T , it follows that

∂T

∂σ2v

= 1σ2v

[1 + T − eTeT − 1

]

Plugging this value back onto ∂Σt/∂σ2v , we get that for t < T ,

∂Σt

∂σ2v

= t − (1 − e−t)[

T

1 − e−T

]≤ 0

The inequality follows from the fact that t/(1 − e−t) is an increasing functionof t.

Let us now prove the monotonicity of the insider’s ex ante expected payoff.First of all, from the expressions for Σt , αt , and γt in Theorem 2, it follows thatE[Πt] = αtΣt + γt is equal to (under the normalization 2μ= 1)

E[Πt] = 2σyσv cosh(

12(T − t)+

)for t ≥ 0

Note that to prove the monotonicity of E[Πt] with respect to σv, it is enoughto focus on the case t ≤ T . The derivative with respect to σv is given by

∂E[Πt]∂σv

= 2σy cosh(

12(T − t)

)+ σyσv sinh

(12(T − t)

)∂T

∂σv

= 2σy cosh(

12(T − t)

)+ 2σy sinh

(12(T − t)

)[1 + T − eTeT − 1

]


= 2σy sinh(

12(T − t)

)[T

eT − 1

]+ 2σy exp

(T − t

2

)≥ 0 Q.E.D.

PROOF OF THEOREM 3: The insider’s Hamilton–Jacobi–Bellman (HJB) op-timality condition are given by

0 = maxβ

{−λtβMΠM + 1

2λ2t σ

2yΠMM +Πt −μΠ +M2β

}for t ∈ [0�∞)

Suppose, we guess a quadratic value function of the formΠ(t�M)= αtM2 +γtfor deterministic functions αt and γt . The HJB equation is satisfied if and onlyif αt − μαt = 0, 1 − 2λtαt = 0, and αt(σ2

v (t)+ λ2t σ

2y )+ γt − μγt = 0. The first

two conditions lead to λt = λ0e−μt and αt = eμt/[2λ0] for some constant λ0 > 0.

Replacing these two functions, the solution of the last differential equation is

γt = 12λ0

(C + Γt)eμt +σ2y λt

4μ

for some constant C ≥ 0 (since Γt ↓ 0 as t → ∞, C ≥ 0 is required to ensurethat γt ≥ 0 for all t).

Note that the HJB condition does not provide any information about how toselect the insider’s strategy βt . (Effectively, we have solved the HJB equationusing the fact that the insider is indeed indifferent.) To determine the valueof βt , we must turn to the market maker’s filtering conditions. The conditionPt = E[Vt|FM

t ] implies that Pt is the orthogonal projection Vt on FMt in L2,

and we can interpret the equilibrium market price as the solution to a classicalKalman–Bucy filtering problem. Let the signal process be the value of the fun-damental Vt , with dynamics dVt = σv dB

vt and the observation process be the

price process Pt , with dynamics dPt = λt dZt = βtλt(Vt −Pt)dt+σyλt dByt . Letvt be the corresponding optimal (in mean square sense) filtering estimate of Vtand let Σt be the filtering error. Then the equilibrium condition is Pt = vt . Thegeneralized Kalman filter conditions for the pair (Vt�Pt) are given by

dvt = Σtβt

λtσ2y

[dPt − λtβt(vt − Pt)dt]� Σt = σ2v − (Σtβt)

2

σ2y

To recover the identity Pt = vt , we need to impose that Σtβt = λtσ2y . This equal-

ity together with the border condition v0 = P0 imply that vt = Pt for all t > 0.This equality also implies that (Σtβt)2 = λ2

t σ4y . Therefore, the market maker’s

filtering conditions are

Σtβt = λtσ2y � Σt = σ2

v (t)− σ2y λ

2t �


which guarantee that the market maker equilibrium condition Pt = E[Vt|FMt ]

is satisfied. Since λt = λ0e−μt , it follows that

Σt = Σ0 + Γ0 − Γt −σ2y λ

20

2μ(1 − e−μt)� βt =

σ2y λ0e

−μt

Σt

To complete the proof, we need to specify the values of the two constantλ0 and C and verify that the proposed value function Π(t�M) = αtM

2 + γtand trading strategy βt effectively solve the insider’s problem. This final step isachieved by imposing the transversality condition limt→∞ e−μt

E[Π(t�Mt)] = 0for βt .

To avoid confusion, we now use βt to denote the trading strategy in equa-tion (10) use and βt to denote an arbitrary policy in B. To explicate the de-pendence of Mt on a trading strategy {βs : 0 ≤ s ≤ t}, we will use the notationMt(β).

SinceΠ(t�M)= αtM2 +γt satisfies the HJB equation for any strategy β ∈ B,it follows that

Π(0�M0)= E

[∫ t

0e−μsβsM2

s (β)ds+ e−μtΠ(t�Mt(β))

](22)

≥ E

[∫ t

0e−μsβsM2

s (β)ds

]

In addition,

e−μtE[Π(t�Mt(βt))

]= 12λ0

(E[M2

t (β)] +C + Γt +σ2y λ

20e

−2μt

2μ

)

Thus, the transversality condition holds only ifC = 0 and limt→∞ E[M2t (β)] = 0.

We can show that

E[M2t (β)] =M2

0e−2λ0

∫ t0 e

−μsβs ds

+∫ t

0e−2λ0

∫ ts e

−μuβu du(σ2v (s)+ σ2

y λ20e

−2μs)ds

Hence, limt→∞ E[M2t (β)] = 0 if

∫ ∞0 e−μsβs ds = ∞. This last requirement to-

gether with the fact that βt = σ2y λ0e

−μt/Σt imply that limt→∞Σt = 0. Therefore,

λ0 =√

2μ(Σ0 + Γ0)

σ2y


With these choices of λ0 and C, the transversality condition is satisfied for βt ,and taking limits in equation (22), we get that

Π(0�M0)= E

[∫ ∞

0e−μsβsM2

s (β)ds

]≥ E

[∫ ∞

0e−μsβsM2

s (β)ds

]for all β ∈ B

In the last step, we used (9) and βt > 0 for all t, and invoked the Lebesgueconvergence theorem to interchange limits and expectations.

We conclude the proof by showing that the equilibrium strategy satisfies con-dition (9). Given the expression for E[Mt] above, this condition is equivalentto

M20

∫ ∞

0fte

−ft dt +∫ ∞

0fte

−ft[∫ t

0(σ2

v (s)+ σ2y λ

20e

−2μs)efs ds

]dt <∞�

where ft := 2λ0

∫ t0 e

−μsβs ds and ft is its first derivative with respect to t. Notethat the first integral is equal to 1. Using the Fubini theorem to reverse theorder of integration, the second integral is equal to∫ ∞

0(σ2

v (s)+ σ2y λ

20e

−2μs)efs[∫ ∞

s

fte−ft dt

]ds

=∫ ∞

0(σ2

v (s)+ σ2y λ

20e

−2μs)ds= Γ0 + σ2y λ

20/(2μ) <∞ Q.E.D.

REFERENCES

AMIHUD, Y., H. MENDELSON, AND L. PEDERSEN (2006): “Liquidity and Asset Prices,” Founda-tions and Trends in Finance, 1, 269–364. [248]

BACK, K. (1992): “Insider Trading in Continuous Time,” Review of Financial Studies, 5, 387–409.[246,259]

BACK, K., AND S. BARUCH (2004): “Information in Securities Markets: Kyle Meets Glosten andMilgrom,” Econometrica, 72, 433–465. [246]

BACK, K., AND H. PEDERSEN (1998): “Long-Lived Information and Intraday Patterns,” Journalof Financial Markets, 1, 385–402. [248]

BAGEHOT, W. (1971): “The Only Game in Town,” Financial Analyst Journal, 22, 12–14. [245]BIAIS, B., L. GLOSTEN, AND C. SPATT (2005): “Market Microstructure: A Survey of Microfun-

dations, Empirical Results and Policy Implications,” Journal of Financial Markets, 8, 217–264.[248]

BRUNNERMEIER, M. (2001): Asset Pricing Under Asymmetric Information. New York: Oxford Uni-versity Press. [248]

CHAU, M., AND D. VAYANOS (2008): “Strong-Form Efficiency With Monopolistic Insiders,” TheReview of Financial Studies, 21, 2275–2306. [247,248]

FOSTER, F., AND S. VISWANATHAN (1996): “Strategic Trading When Agents Forecast the Fore-casts of Others,” The Journal of Finance, 51, 1437–1478. [248]

http://www.e-publications.org/srv/ecta/linkserver/setprefs?rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:1/AmihudSurvey&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:2/Back&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:3/Baruck&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:4/BackPedersen&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:5/Bagehot&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:6/Biais&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:8/Chau&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:9/FosterViswanathan&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:1/AmihudSurvey&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:3/Baruck&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:4/BackPedersen&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:6/Biais&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:8/Chau&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:9/FosterViswanathan&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8


GLOSTEN, L., AND P. MILGROM (1985): “Bid, Ask and Transaction Prices in a Specialist MarketWith Heterogeneously Informed Traders,” Journal of Financial Economics, 14, 71–100. [245,246]

HOLDEN, G., AND A. SUBRAHMANYAM (1992): “Long-Lived Private Information and ImperfectCompetition,” The Journal of Finance, 47, 247–270. [248]

KNOPP, K. (1990): Theory and Applications of Infinite Series. New York: Dover. [271]KYLE, A. (1985): “Continuous Auctions and Insider Trading,” Econometrica, 53, 1315–1335. [245,

246,250,257,258]MENDELSON, H., AND T. TUNCA (2004): “Strategic Trading, Liquidity, and Information Acquisi-

tion,” Review of Financial Studies, 17, 295–337. [248]O’HARA, M. (1997): Market Microstucture Theory. MA: Blackwell Publishing. [248]PROKHOROV, Y. (1956): “Convergence of Random Processes and Limit Theorems in Probability

Theory,” Theory of Probability and It’s Applications, 1, 157–214. [274]SPIEGEL, M., AND A. SUBRAHMANYAM (1992): “Informed Speculation and Hedging in a Non-

competitive Securities Market,” Review of Financial Studies, 5, 307–329. [248]

Stern School of Business, New York University, 44 West Fourth Street, Suite 8-77,New York, NY 10012, U.S.A.; [email protected]

andDept. of Economics, New York University, 19 West Fourth Street, 6th Floor, New

York, NY 10012, U.S.A.; [email protected].

Manuscript received April, 2008; final revision received July, 2009.

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:10/Glosten&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:11/HoldenSubrahmanyam92&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:13/Kyle&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:14/Tunca&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:16/Prokhorov56&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:17/SpiegelSubrahmanyam92&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

mailto:[email protected]

mailto:[email protected]

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:10/Glosten&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:11/HoldenSubrahmanyam92&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:14/Tunca&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:16/Prokhorov56&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

http://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:17/SpiegelSubrahmanyam92&rfe_id=urn:sici%2F0012-9682%28201001%2978%3A1%3C245%3AITWARD%3E2.0.CO%3B2-8

Documents

Insider Trading With a Random Deadlinepeople.stern.nyu.edu/rcaldent/papers/RInsider.pdf · insider could let the information asymmetry (the variance of the uninformed traders’ estimate